Method and system for determining an association of biological features with a medical condition

ABSTRACT

A system and a method for determining an association of one or more biological features with a medical condition provides empirical results and simulations confirming that the involvement of both L 1/2 -regularized logistic regression and L 2 -regularized logistic regression in the regression model is highly competitive against usual approaches like Lasso, L 1/2 , SCAD-L 2 , and Elastic net in analyzing high dimensional and low sample sizes data.

TECHNICAL FIELD

The present invention relates to a system and a method for determining an association of one or more biological features with a medical condition, in particularly, but not exclusively, to a system for determining an association of one or more of presence of genes, gene expression, presence of gene products or amount of gene products with cancer, in particular for determining an association of the expression of one or more genes with cancer or a specific cancer subtype.

BACKGROUND

Cancer classification and feature (gene) selection plays an important role in knowledge discovery in genomic data.

With advances in high-throughput molecular techniques, researchers can study the expression of tens thousands of genes simultaneously. Cancer classification based on gene expression levels is one of the central problems in genome research. Logistic regression is a popular classification method and has an explicit statistical interpretation which can obtain probabilities of classification regarding the cancer phenotype. However, in most gene expression studies, the number of genes typically far exceeds the number of sample sizes. This situation is called high dimensional and low sample size problem and the normal logistic regression method cannot be directly used to obtain the parameter estimates. Thus, although logistic regression is one of the most popular classification methods, it does unfortunately not achieve the purpose of the feature selection.

There remains a strong need for methods and related systems sufficiently dealing with the problem of highly correlated variables and the high dimensional and low sample size problem, i.e. those which are suitable for feature selection in particular in the medical field. Clearly, having a fast and reliable analysis of features could significantly contribute to an improved diagnosis of diseases and treatment selection.

SUMMARY OF THE INVENTION

In accordance with a first aspect of the present invention, there is provided a method for determining an association of one or more biological features with a medical condition, comprising the steps of:

obtaining a dataset comprising biological data related to a plurality of samples each having a plurality of biological features;

applying at least some of the biological data to a regression model to determine and/or optimize parameters in the regression model thereby solving the regression model; and

processing the biological data using the solved regression model to determine one or more biological features that are associated with the medical condition;

wherein the regression model involves the use of both L_(1/2)-regularized logistic regression and L₂-regularized logistic regression.

The expression “association of one or more biological features with a medical condition” generally means any kind of expected link or connection between both, in an embodiment “association” means that the medical condition is or is expected to be

a) caused by the one or more biological features,

b) the one or more biological features at least contribute to the medical condition or its clinical worsening, and/or

c) the one or more biological features are involved in cellular processes which cause or contribute or are expected to cause or contribute to the medical condition or its clinical worsening.

In an embodiment, the regression model involves the use of a linear combination of L_(1/2)-regularized logistic regression and L₂-regularized logistic regression.

In an embodiment, the dataset is represented as:

D={(X ₁ , y ₁), (X ₂ , y ₂), . . . , (X _(n) , y _(n))},

where n is a number of the plurality of samples, p is a number of the plurality of biological features, X_(i)=(x_(i1),x_(i2), . . . ,x_(ip)) is the i^(th) sample with p biological features, and y_(i) is a corresponding dependent variable. The regression model is in an embodiment, in particular in the afore-mentioned embodiment, defined at least partly by:

${L\left( {\lambda,\alpha,\beta} \right)} = {{- {\sum\limits_{i = 1}^{n}\left\{ {{y_{i}{\log \left\lbrack {f\left( {X_{i}^{\prime}\beta} \right)} \right\rbrack}} + {\left( {1 - y_{i}} \right){\log \left\lbrack {1 - {f\left( {X_{i}^{\prime}\beta} \right)}} \right\rbrack}}} \right\}}} + {\lambda\left( {{\alpha {\beta }^{\frac{1}{2}}} + {\left( {1 - \alpha} \right){\beta }^{2}}} \right)}}$

where β=(β₁, . . . , β_(p)) are the coefficients to be determined and/or optimized, and λ and α are predetermined values. In the afore-mentioned embodiment, in particular λ=1 and α=0.2.

In an embodiment, such as in an afore-mentioned embodiment, the parameters in the regression model are determined and/or optimized using a coordinate descent optimization method. Said coordinate descent optimization method in a further embodiment comprises the steps of:

optimizing the regression model using at least some of the biological data, by varying one of the parameters in the regression model while fixing the other parameters in the regression model;

updating the parameters in the regression model; and

repeating the optimization and update steps for at least once for each of the parameters in the regression model so as to optimize the regression model.

The repeating step continues in an embodiment until a convergence condition relating to the parameters in the regression model is reached.

In an embodiment, the method of the present invention further comprises the step of ranking the biological features determined to be associated with the medical condition according to their degree of association with the medical condition.

In an embodiment, the plurality of samples according to the method of the present invention is taken from animals or humans, at least some of which have the medical condition. In a further embodiment, the plurality of samples is taken from mammals and in particular from humans, at least some of which have the medical condition.

In an embodiment, the samples are of human tissue like cancerous tissue.

In an embodiment, the medical condition includes and in particular is cancer.

In an embodiment, the plurality of biological features is one or more of: presence of a gene, gene expression, presence of a gene product or amount of a gene product. In a particular embodiment, the at least one gene represents a biomarker such as for cancer or a specific cancer subtype.

In accordance with a second aspect of the present invention, there is provided a system for determining an association of one or more biological features with a medical condition, comprising:

one or more processors arranged to

apply at least some biological data in a dataset comprising biological data related to a plurality of samples each having a plurality of biological features to a regression model so as to determine and/or optimize parameters in the regression model thereby solving the regression model; and

process the biological data using the solved regression model so as to determine one or more biological features that are associated with the medical condition;

wherein the regression model involves the use of both L_(1/2)-regularized logistic regression and L₂-regularized logistic regression.

In an embodiment of the system of the present invention, the regression model involves the use of a linear combination of L_(1/2)-regularized logistic regression and L₂-regularized logistic regression.

In an embodiment of the system of the present invention, the dataset is represented as:

D={(X ₁ , y ₁), (X ₂ , y ₂), . . . , (X _(n) , y _(n))},

where n is a number of the plurality of samples, p is a number of the plurality of biological features, X_(i)=(x_(i1),x_(i2), . . . ,x_(ip)) is the i^(th) sample with p biological features, and y_(i) is a corresponding dependent variable. The regression model is in an embodiment of the system of the present invention, in particular in the afore-mentioned embodiment, defined at least partly by:

${L\left( {\lambda,\alpha,\beta} \right)} = {{- {\sum\limits_{i = 1}^{n}\left\{ {{y_{i}{\log \left\lbrack {f\left( {X_{i}^{\prime}\beta} \right)} \right\rbrack}} + {\left( {1 - y_{i}} \right){\log \left\lbrack {1 - {f\left( {X_{i}^{\prime}\beta} \right)}} \right\rbrack}}} \right\}}} + {\lambda\left( {{\alpha {\beta }^{\frac{1}{2}}} + {\left( {1 - \alpha} \right){\beta }^{2}}} \right)}}$

where β=(β₁, . . . , β_(p)) are the coefficients to be optimized, and λ and α are predetermined values. In the afore-mentioned embodiment, in particular λ=1 and α=0.2.

In an embodiment of the system of the present invention, the one or more processors are further arranged to determine and/or optimize the parameters in the regression model by

optimizing the regression model using at least some of the data, through varying one of the parameters in the regression model while fixing the other parameters in the regression model;

updating the parameters in the regression model; and

repeating the optimization and update steps for at least once for each of the parameters in the regression model until a convergence condition relating to the parameters in the regression model is reached.

In an embodiment of the system of the present invention, the one or more processors are further arranged to rank the biological features determined to be associated with the medical condition according to their degree of association with the medical condition.

In an embodiment of the system of the present invention, the plurality of samples is taken from animals or humans, at least some of which have the medical condition. In further embodiments, the plurality of samples is taken from mammals and in particular from humans, at least some of which have the medical condition.

In an embodiment of the system of the present invention, the samples are of human tissue like cancerous tissue.

In an embodiment of the system of the present invention, the medical condition includes and in a particular embodiment is cancer.

In an embodiment of the system of the present invention, the plurality of biological features is one or more of: presence of a gene, gene expression, presence of a gene product or amount of a gene product. In a particular embodiment, the at least one gene represents a biomarker such as for cancer or a specific cancer subtype.

In an embodiment, the system of the present invention further comprises a display arranged to display the determination result.

In accordance with a third aspect of the present invention, there is provided a non-transient computer readable medium for storing computer instructions that, when executed by one or more processors, causes the one or more processors to perform a method for determining an association of one or more biological features with a medical condition, comprising the steps of:

applying at least some biological data in a dataset comprising biological data related to a plurality of samples each having a plurality of biological features to a regression model so as to determine and/or optimize parameters in the regression model thereby solving the regression model; and

processing the biological data using the solved regression model so as to determine one or more biological features that are associated with the medical condition;

wherein the regression model involves the use of both L_(1/2)-regularized logistic regression and L₂-regularized logistic regression.

In an embodiment of the non-transient computer readable medium of the present invention, the regression model involves the use of a linear combination of L_(1/2)-regularized logistic regression and L₂-regularized logistic regression.

In an embodiment of the non-transient computer readable medium of the present invention, the dataset is represented as:

D={(X ₁ , y ₁), (X ₂ , y ₂), . . . , (X _(n) , y _(n))},

where n is a number of the plurality of samples, p is a number of the plurality of biological features, X_(i)=(x_(i1),x_(i2), . . . ,x_(ip)) is the i^(th) sample with p biological features, and y_(i) is a corresponding dependent variable. In an embodiment, in particular in the afore-mentioned embodiment, the regression model is defined at least partly by:

${L\left( {\lambda,\alpha,\beta} \right)} = {{- {\sum\limits_{i = 1}^{n}\left\{ {{y_{i}{\log \left\lbrack {f\left( {X_{i}^{\prime}\beta} \right)} \right\rbrack}} + {\left( {1 - y_{i}} \right){\log \left\lbrack {1 - {f\left( {X_{i}^{\prime}\beta} \right)}} \right\rbrack}}} \right\}}} + {\lambda\left( {{\alpha {\beta }^{\frac{1}{2}}} + {\left( {1 - \alpha} \right){\beta }^{2}}} \right)}}$

where β=(β₁, . . . , β_(p)) are the coefficients to be optimized, and λ and α are predetermined values. In the afore-mentioned embodiment, in particular λ=1 and α=0.2.

In an embodiment of the non-transient computer readable medium of the present invention, the one or more processors are further arranged to determine and/or optimize the parameters in the regression model by

optimizing the regression model using at least some of the data, through varying one of the parameters in the regression model while fixing the other parameters in the regression model;

updating the parameters in the regression model; and

repeating the optimization and update steps for at least once for each of the parameters in the regression model until a convergence condition relating to the parameters in the regression model is reached.

In an embodiment, the one or more processors can be further arranged to rank the biological features determined to be associated with the medical condition according to their degree of association with the medical condition.

In an embodiment of the non-transient computer readable medium of the present invention, the plurality of samples are taken from animals or humans. In further embodiments, the plurality of samples is taken from mammals and in particular from humans at least some of which have the medical condition.

In an embodiment, the samples are of human tissue like cancerous tissue.

The medical condition in an embodiment includes and in a further embodiment is cancer.

The plurality of biological features is in an embodiment of the non-transient computer readable medium one or more of: presence of a gene, gene expression, presence of a gene product or amount of a gene product. In a particular embodiment, the at least one gene represents a biomarker such as for cancer or a specific cancer subtype.

Other features and aspects of the invention will become apparent by consideration of the following detailed description and accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of a computer or computing server arranged to operate a system of the present invention for determining an association of one or more biological features with a medical condition.

FIG. 2 is a schematic diagram showing a system of the present invention for determining an association of one or more biological features with a medical condition.

FIGS. 3A, 3B, 3C, and 3D show exact solutions of Lasso, L_(1/2) Elastic net, and HLX in an orthogonal design. The regularization parameters are λ=0.1 and α=0.8 for Elastic net and HLX (β-OLS is the ordinary least-squares (OLS) estimator). FIG. 3A shows exact solutions of Lasso, FIG. 3B of L_(1/2), FIG. 3C of Elastic net, and FIG. 3D of HLX in an orthogonal design.

FIG. 4 refers to contour plots (two dimensional) for the regularization methods. The regularization parameters are λ=1 and α=0.2 for the HLX method.

FIG. 5 illustrates the performance of the AUC from ROC analyses of each method on every dataset of example 2.

DETAILED DESCRIPTION OF THE INVENTION

The inventors based on their research, tests and experiments concluded that a regularization method can be used in order to deal with the problem of high dimensionality. A well-known regularization method is the L₁ penalty (Tibshirani R., J R Stat Soc B. JSTOR; 1996; 267-288), which is the least absolute shrinkage and selection operator (Lasso). It is performing continuous shrinkage and gene selection at the same time. Other L₁ norm type regularization methods include the smoothly-clipped-absolute-deviation (SCAD) penalty (Fan J., Li R., J Am Stat Assoc. Taylor & Francis; 2001;96: 1348-1360) which is symmetric, nonconcave, and has singularities at the origin to produce sparse solutions. The adaptive Lasso (Zou H., J Am Stat Assoc. Taylor & Francis; 2006;101: 1418-1429) where adaptive weights are employed to penalize different coefficients in the L₁ penalty. However, the L₁ type regularization may yield inconsistent feature selections in some situations (Zou H., J Am Stat Assoc; 2006;101: 1418-1429) and often introduces extra bias in estimation (Meinshausen N., Yu B., Ann Stat. JSTOR; 2009; 246-270).

Xu et al. (Xu Z. et al., Sci China Inf Sci. Springer; 2010;53: 1159-1169) proposed the L_(1/2) penalty, a method that can be taken as a representative of L_(q) (0<q<1) penalties in both sparsity and computational efficiency, and has demonstrate many attractive properties, such as unbiasedness, and oracle properties (Xu Z. et al. , Sci China Inf Sci. Springer; 2010;53: 1159-1169, Zeng J. et al., Signal Process IEEE Trans. IEEE; 2014;62: 2317-2329, Xu Z. et al., IEEE Trans neural networks Learn Syst. United States; 2012;23: 1013-1027, doi:10.1109/TNNLS.2012.2197412 [doi]). However, similar to most of regularization methods, the L_(1/2) penalty ignores the correlation between features, and is consequently unable to analyze data with dependent structures. If there is a group of variables among which the pair-wise correlations are very high, then the L_(1/2) method tends to select only one variable that represents the corresponding group.

In gene expression study, genes are often highly correlated if they share the same biological pathway (Segal M R. et al, J Comput Biol. Mary Ann Liebert, Inc.; 2003;10: 961-980). Some efforts had been made to deal with the problem of highly correlated variables. Zhou and Hastie proposed Elastic net penalty (Zou H, Hastie T., J R Stat Soc Ser B., Wiley Online Library; 2005;67: 301-320) which is a linear combination of L₁ and L₂ (the ridge technique) penalties, and such method emphasizes a grouping effect, where strongly correlated genes tend to be in or out of the model together. Becker et al. (Becker N. et al., BMC Bioinformatics. German Cancer Research Center, Division Molecular Genetics, INF 280, 69120 Heidelberg; 2011;12: 138. doi:10.1186/1471-2105-12-138 [doi]) proposed the Elastic SCAD (SCAD-L₂), a combination of SCAD and L₂ penalties. By introducing the L₂ penalty term, Elastic SCAD also works for the groups of predictors.

Without being bound by theory, the inventors herein through their research, tests and experiments discovered that a HLX approach with involvement of both, L_(1/2)-regularized logistic regression and L₂-regularized logistic regression in the regression model, can be advantageously applied when determining an association of biological features with a medical condition.

In this embodiment, the system for determining an association of biological features with a medical condition is implemented by or for operation on a computer having an appropriate user interface. The computer may be implemented by any computing architecture, including stand-alone PC, client/server architecture, “dumb” terminal/mainframe architecture, or any other appropriate architecture. The computing device is appropriately programmed to implement the invention.

Referring to FIG. 1, there is a shown a schematic diagram of a computer or a computing server 100 which in this embodiment comprises a server 100 arranged to operate, at least in part if not entirely, the system for determining an association of one or more biological features with a medical condition in accordance with one embodiment of the present invention. The server 100 comprises suitable components necessary to receive, store and execute appropriate computer instructions. The components may include a processing unit 102, read-only memory (ROM) 104, random access memory (RAM) 106, and input/output devices such as disk drives 108, input devices 110 such as an Ethernet port, a USB port, etc., display 112 such as a liquid crystal display, a light emitting display or any other suitable display and communications links 114. The server 100 includes instructions that may be included in ROM 104, RAM 106 or disk drives 108 and may be executed by the processing unit 102. There may be provided a plurality of communication links 114 which may variously connect to one or more computing devices such as a server, personal computers, terminals, wireless or handheld computing devices. At least one of a plurality of communications link may be connected to an external computing network through a telephone line or other type of communications link.

The server 100 may include storage devices such as a disk drive 108 which may encompass solid state drives, hard disk drives, optical drives or magnetic tape drives. The server 100 may use a single disk drive or multiple disk drives. The server 100 may also have a suitable operating system 116 which resides on the disk drive or in the ROM of the server 100.

The system has a database 120 residing on a disk or other storage device which is arranged to store a dataset. The database 120 is in communication with the server 100 with an interface, which is implemented by computer software residing on the server 100. Alternatively, the database 120 may also be implemented as a stand-alone database system in communication with the server 100 via an external computing network, or other types of communication links.

With reference to FIG. 2, there is provided a system for determining an association of biological features with a medical condition, comprising:

one or more processors 206 arranged to

apply at least some biological data in a dataset (200) comprising biological data related to a plurality of samples each having a plurality of biological features to a regression model (202) so as to determine and/or optimize parameters in the regression model thereby solving the regression model; and

process the biological data using the solved regression model (204) so as to determine one or more biological features (208) that are associated with the medical condition;

wherein the regression model involves the use of both L_(1/2)-regularized logistic regression and L₂-regularized logistic regression.

In this embodiment, the system may include one or more processors (206) each arranged to apply at least some biological data in a dataset (200) comprising biological data related to a plurality of samples from cancerous tissue from humans each having a plurality of biological features, namely presence of a gene, gene expression, presence of gene product and/or amount of gene product, to a regression model (202) to determine and/or optimize parameters in the regression model thereby solving the regression model.

The regression model involves the HLX penalty as a combination of L_(1/2) and L₂ penalties. Namely, the regression model can involve in this embodiment a linear combination of both L_(1/2)-regularized logistic regression and L₂-regularized logistic regression. The one or more processors (206) are further each arranged to process the biological data using the solved regression model (204) to determine one or more biological features (208) that are associated with the medical condition such as presence of one or more gene and its association with cancer or a specific subtype of cancer.

These processes, which can include methods of the present invention, may be implemented as a plurality of steps on a computer or computing device, such as those as found in FIG. 1.

Empirical results and simulations confirm that the system of the present invention provides several advantages over known approaches for feature selection. More specifically, empirical results and simulations show that the HLX regularization with involvement of both, L_(1/2)-regularized logistic regression and L₂-regularized logistic regression in the regression model, is highly competitive amongst usual approaches like Lasso, L_(1/2) , SCAD-L₂ and Elastic net in analyzing high dimensional and low sample sizes data like microarray and RNS-seq. data.

Thus, logistic regression with the HLX method represents a promising tool for feature selection in the classification problem.

The system of the present invention is in particular characterized by a new and advantageous HLX penalized function, which is in an embodiment a linear combination of L_(1/2) and L₂ penalties, such as to select the relevant gene in the logistic regression. The HLX method inherits advantageous characteristics from the L_(1/2) (sparsity) and L₂ (grouping effect where highly correlated variables are in or out a model together) penalties. Further, a novel univariate HLX thresholding approach is provided in embodiments to update the estimated coefficients and a coordinate descent process for the HLX penalized logistic regression model.

More specifically, a strictly convex penalty function provides a sufficient condition for the grouping effect of variables and the L₂ penalty guarantees strict convexity (Zeng L, Xie J, Statistics (Ber)., Taylor & Francis; 2014;48: 49-66). Therefore, in the HLX regularization, the L_(1/2) penalty advantageously achieves feature selection, and the L₂ penalty induces the grouping effect simultaneously.

Preferably but not exclusively, the system of the present invention can be used for determining an association of one or more of presence of a gene or genes, respective gene(s) expression, presence of a gene product or gene products and/or respective expression of gene product(s) with a medical condition like cancer or a specific subtype of cancer. In particular, the system of the present invention can be utilized for analyzing a dataset with biological data from humans including a plurality of gene expression levels.

Further features, applications and advantages of the method, the system and the non-transient computer readable medium for storing computer instructions according to the present invention will be evident for a person skilled in the art from the features and embodiments described below relating to a HLX regularization and a method for solving the logistic regression model.

In this embodiment, it is supposed that X=[x₁|, . . . , |x_(p)] is the matrix of n samples with p genes, x_(j)=(x_(1j), . . . ,x_(nj))^(T) for j=1, . . . ,p, and y=(y₁, . . . ,y_(n))^(T) is the corresponding response. For any non-negative λ, the normal regularization form is:

$\begin{matrix} {{L\left( {\lambda,\beta} \right)} = {{\arg \; \min \frac{1}{n}{\sum\limits_{i = 1}^{n}\left( {y - {X^{\prime}\beta}} \right)^{2}}} + {\lambda \; {P(\beta)}}}} & (1) \end{matrix}$

where P(β) represents the regularization term. As mentioned before, there are many regularization methods proposed in recent years. One of popular method is the already described L₁ regularization (Lasso), where

P(β)=Σ_(j=1) ^(p)|β_(j)|¹.

The other L₁ type regularizations include SCAD, the adaptive Lasso, Elastic net, Stage wise Lasso (Zhao P, Yu B., J Mach Learn Res. JMLR. org; 2007;8: 2701-2726), Dantzig selector (Candes E, Tao T., Ann Stat. JSTOR; 2007; 2313-2351) and Elastic SCAD. However, in genomic research, the result of the L₁type regularization may not sparse enough for interpretation. Actually, a typical microarray or RNA-seq data set has many thousands of predictors (genes), researchers often desire to select fewer but informative genes. Beside this, the L₁ regularization is asymptotically biased (Knight K, Fu W., Ann Stat. JSTOR; 2000; 1356-1378, Malioutov D. et al., Signal Process IEEE Trans. IEEE; 2005;53: 3010-3022). Although the L₀ regularization, where

P(β)=Σ_(j=1) ^(p)|β_(j)|⁰,

yields the sparsest solutions, it has to deal with a NP hard combinatory optimization problem. To gain more concise solution and improve its predictive accuracy, one needs to think beyond the L₁and L₀ regularization to the L_(q) (0<q<1) regularization.

The L_(1/2) regularization can be taken as a representative of the L_(q) (0<q<1) penalties and has permitted an analytically expressive thresholding representation (Xu Z. et al., Sci China Inf Sci. Springer; 2010;53: 1159-1169). With the thresholding representation, solving the L_(1/2) regularization is much easier than solving the L₀ regularization. Moreover, the L_(1/2) penalty is unbiasedness and has oracle properties (Xu Z. et al., Sci China Inf Sci. Springer; 2010;53: 1159-1169, Zeng J. et al., Signal Process IEEE Trans. IEEE; 2014;62: 2317-2329, Xu Z. et al., IEEE Trans neural networks Learn Syst. United States; 2012;23: 1013-1027, doi:10.1109/TNNLS.2012.2197412 [doi]).

These characteristics are making the L_(1/2) penalty became an efficient tool for high dimensional problem (Liang Y. et al., BMC Bioinformatics. Faculty of Information Technology & State Key Laboratory of Quality Research in Chinese Medicines, Macau University of Science and Technology, Macau, China; 2013;14: 198. doi:10.1186/1471-2105-14-198 [doi], Huang H-H. et al., Biomed Res Int. Hindawi Publishing Corporation; 2015). However, due to the insensitivity of the highly correlated data, the L_(1/2) penalty tends to select only one variable represents the correlated group. This drawback may deteriorate the performance of the L_(1/2) method.

For any fixed non-negative λ₁ and λ₂, the HLX criterion is defined:

$\begin{matrix} {{L\left( {\lambda_{1},\lambda_{2},\beta} \right)} = {{\arg \; \min \frac{1}{n}{\sum\limits_{i = 1}^{n}\left( {y - {X^{\prime}\beta}} \right)^{2}}} + {\lambda_{1}{\beta }^{1/2}} + {\lambda_{2}{\beta }^{2}}}} & (2) \end{matrix}$

where β=(β₁, . . . , β_(p)) are the coefficients to be estimated and

|β|^(1/2)=Σ_(j=1) ^(p)|β_(j)|^(1/2),

|β|²=Σ_(j=1) ^(p)|β_(j)|².

The HLX estimator {circumflex over (β)} is the minimizer of Equation (2):

{circumflex over (β)}=arg min_(β) {L(λ₁,λ₂,β)}.   (3)

Let α=λ₁/(1+λ₂), then solving {circumflex over (β)} in Equation (3) is equivalent to the optimization problem:

{circumflex over (β)}=arg min_(β){|y−X′β|²+λ(α|β|^(1/2)+(1−α)|β|²)}  (4)

The function α|β|^(1/2)+(1−α)|β|² is called the HLX penalty, which is a combination of the L_(1/2) and L₂ penalties. When α=0, the HLX penalty becomes ridge regularization. When α=1, the HLX penalty becomes L_(1/2) regularization. The L₂ penalty is enjoying the grouping effect and the L_(1/2) induces sparse solutions. This penalty combination makes the HLX regularization not only capable of dealing with the correlation data but also able to generate a succinct result.

In FIGS. 3A to 3D four regularization methods are shown: Lasso, L_(1/2), Elastic net and HLX penalties with an orthogonal design matrix in the regression model. The estimators of Lasso and Elastic net are biased, whereas L_(1/2) is asymptotically unbiased. Similar to L_(1/2), the new HLX approach also performs better than Lasso and Elastic net in the property of unbiasedness.

In FIG. 4 the contour plots on two dimensional for the penalty functions of Lasso, Elastic net, L_(1/2) and new HLX approaches are described. It is clear that the L_(1/2) penalty is non-convex, whereas the HLX penalty is convex for the given α. The following theorem will show how the HLX regularization strengthens the L_(1/2) regularization.

Theorem 1. Given dataset (y, X) and (λ₁,λ₂), then the HLX estimates {circumflex over (β)} are given by

$\begin{matrix} {\hat{\beta} = {{\arg \; {\min_{\beta}{{\beta^{T}\left( \frac{{X^{T}X} + {\lambda_{2}I}}{1 + \lambda_{2}} \right)}\beta}}} - {2y^{T}X\; \beta} + {\lambda_{1}{{\beta }^{2}.}}}} & (5) \end{matrix}$

The L_(1/2) regularization can be rewritten as

{circumflex over (β)}(L _(1/2))=arg min_(β)β^(T)(X ^(T) X)β−2y ^(T) Xβ+λ ₁|β|^(1/2).   (6)

The proof of theorem 1 is similar to that in Zou and Hastie (J R Stat Soo Ser B, Statistical Methodol. Wiley Online Library; 2005;67: 301-320).

Lemma 1. Given dataset (y, X) and (λ₁, λ₂), define an artificial dataset (y*, X*) by

${X_{{({n + p})} \times p}^{*} = {\left( {1 + \lambda_{2}} \right)^{- \frac{1}{2}}\left( {\overset{X}{\sqrt{\lambda_{2}}}I} \right)}},{Y_{({n + p})}^{*} = {{{\begin{pmatrix} Y \\ 0 \end{pmatrix}.{Let}}\mspace{14mu} \gamma} = {{{\lambda_{1}/\sqrt{1 + \lambda_{2}}}\mspace{14mu} {and}\mspace{14mu} \beta^{*}} = {\sqrt{1 + \lambda_{2}}\beta}}}}$

Then the HLX criterion can be expressed as)

L(λ₁,λ₂,β)=L(γ,β*)=|y*−X*β*| ²+γ|β*|_(1/2)

Let

β*=arg min_(β) {L(γ,β*)},

Then

{circumflex over (β)}=(1/√{square root over (1+λ₂)}){circumflex over (β)}*.

Assume {circumflex over (β)} is the estimates for the HLX regularization. By Lemma 1 and equation (2) we have

$\begin{matrix} \begin{matrix} {\overset{¨}{\beta} = {{\arg \; {\min_{\beta}{{y^{*} - {X^{*}\frac{\beta}{\sqrt{1 + \lambda_{2}}}}}}^{2}}} + {\frac{\lambda_{1}}{\sqrt{1 + \lambda_{2}}}{\frac{\beta}{\sqrt{1 + \lambda_{2}}}}_{1/2}}}} \\ {= {{\arg \; {\min_{\beta}{{\beta^{T}\left( \frac{X^{*T}X^{*}}{1 + \lambda_{2}} \right)}\beta}}} - {2\frac{y^{*T}X^{*}\beta}{\sqrt{1 + \lambda_{2}}}} + {y^{*T}y^{*}} + {\frac{\lambda_{1}{\beta }_{1/2}}{1 + \lambda_{2}}.}}} \end{matrix} & (13) \end{matrix}$

Substituting the identities

${{X^{*T}X^{*}} = \left( \frac{{X^{T}X} + \lambda_{2}}{1 + \lambda_{2}} \right)},{{y^{*T}X^{*}} = \left( \frac{y^{T}X}{\sqrt{1 + \lambda_{2}}} \right)},{{y^{*T}y^{*}} = {y^{*T}y}}$

into equation (13), we have

$\begin{matrix} {\hat{\beta} = {{\arg \; {\min_{\beta}{\frac{1}{1 + \lambda_{2}}\left\{ {{{\beta^{T}\left( \frac{{X^{T}X} + {\lambda_{2}I}}{1 + \lambda_{2}} \right)}\beta} - {2y^{T}X\; \beta} + {\lambda_{1}{\beta }_{1/2}}} \right\}}}} + {y^{T}y}}} \\ {= {{\arg \; {\min_{\beta}{{\beta^{T}\left( \frac{{X^{T}X} + {\lambda_{2}I}}{1 + \lambda_{2}} \right)}\beta}}} - {2y^{T}X\; \beta} + {\lambda_{1}{{\beta }_{1/2}.}}}} \end{matrix}$

Theorem 1 shows the new HLX approach is a stabilized version of the L_(1/2). Note that {circumflex over (Σ)}=X^(T)X is a sample version of the correlation matrix Σ and

$\frac{{X^{T}X} + {\lambda_{2}I}}{1 + \lambda_{2}} = {{\left( {1 - \delta} \right)} + {\delta \; I}}$

With δ=λ₂/(1+λ₂) shrinks {circumflex over (Σ)} towards the identitymatrix.

The classification accuracy can often be enhanced by replacing {circumflex over (Σ)} by a more shrunken estimate in linear discriminate analysis (Friedman J H., J Am Stat Assoc. Taylor & Francis; 1989;84: 165-175, Hastie T. et al., Math Intell. Springer; 2005;27: 83-85). In other word, the HLX regularization improves the L_(1/2) technique by regularizing {circumflex over (Σ)} in Equation (6).

For sparse Logistic Regression with the HLX regularization it is supposed that dataset D has n samples D={(X₁,y₁), (X ₂, y₂), . . . (X_(n), y_(n))}, where X_(i)=(x_(i1),x_(i2), . . . ,x_(ip)) is i^(th) sample with p genes and y_(i) is the corresponding dependent variable that consist of a binary value with 0 or 1. Define a classifier f(x)=e^(x)/(1+e^(x)) and the logistic regression is defined as:

$\begin{matrix} {{P\left( {y_{i} = {1X_{i}}} \right)} = {{f\left( {X_{i}^{\prime}\beta} \right)} = \frac{\exp \left( {X_{i}^{\prime}\beta} \right)}{1 + {\exp \left( {X_{i}^{\prime}\beta} \right)}}}} & (7) \end{matrix}$

where β=(β₁, . . . , β_(p)) are the coefficients to be estimated. The regression model can, thus, be presented as:

L(β)=−Σ_(i=1) ^(n) {y _(i) log[f(X′ _(i)β)]+(1−y ₁)log[1−f(X′ _(i)β)]}  (8)

In the present invention, HLX penalty is applied to the logistic regression model. For any fixed non-negative λ and α, the sparse logistic regression model based on the HLX regularization is defined in embodiments of the present invention as:

$\begin{matrix} {{L\left( {\lambda,\alpha,\beta} \right)} = {{- {\sum\limits_{i = 1}^{n}\left\{ {{y_{i}{\log \left\lbrack {f\left( {X_{i}^{\prime}\beta} \right)} \right\rbrack}} + {\left( {1 - y_{i}} \right){\log \left\lbrack {1 - {f\left( {X_{i}^{\prime}\beta} \right)}} \right\rbrack}}} \right\}}} + {\lambda \left( {{\alpha {\beta }^{1/2}} + {\left( {1 - \alpha} \right){\beta }^{2}}} \right)}}} & (9) \end{matrix}$

Solution for Logistic Regression with the HLX regularization: The coordinate descent method (Friedman J. et al., J Stat Softw. NIH Public Access; 2010;33: 1) is an efficient method for solving regularization models. The computational time increases linearly with the dimension p, make this method particularly fit for solving high-dimensional problems (Breheny P, Huang J., Ann Appl Stat. NIH Public Access; 2011;5: 232). Its standard procedure can be showed as follows: for every β_(j)(j=1, 2, . . . , p), to partially optimize the target function with respect to coefficient with the remaining elements of β fixed at their most recently updated values, iteratively cycling through all coefficients until meet converged.

The specific form of renewing coefficients is associated with the thresholding operator of the penalty. Following Friedman et al. (Friedman J. et al., J Stat Softw. NIH Public Access; 2010;33: 1) and Liang et al. (Liang Y, et al., BMC Bioinformatics. Faculty of Information Technology & State Key Laboratory of Quality Research in Chinese Medicines, Macau University of Science and Technology, Macau, China; 2013;14: 198, doi:10.1186/1471-2105-14-198 [doi]), in this invention, a coordinate-wise update form for the HLX penalty is presented:

$\begin{matrix} \left. \beta_{j}\leftarrow\frac{{Half}\mspace{14mu} \left( {\omega_{j},{\lambda\alpha}} \right)}{1 + {\lambda \left( {1 - \alpha} \right)}} \right. & (10) \end{matrix}$

Where ω_(j)=Σ_(i=1) ^(n)x_(ij)(y_(i)−{tilde over (y)}_(i) ^((j))) and {tilde over (y)}_(i) ^((j))=Σ_(k≠j)x_(ik)β_(k) as the partial residual for fitting β_(j). Half (

,

) is the L_(1/2) thresholding operator

$\begin{matrix} {{{Half}\mspace{14mu} \left( {\omega_{j},\lambda} \right)} = \left\{ {{{\begin{matrix} {\frac{2}{3}{\omega_{j}\left( {1 + {\cos \left( \frac{2\left( {\pi - {\phi_{\lambda}\left( \omega_{j} \right)}} \right)}{3} \right)}} \right)}} & {{{if}\mspace{14mu} {\omega_{j}}} > {\frac{3}{4}(\lambda)^{\frac{2}{3}}}} \\ 0 & {otherwise} \end{matrix}\mspace{20mu} {where}\mspace{14mu} {\phi_{\lambda}(\omega)}} = {\arccos \left( {\frac{\lambda}{8}\left( \frac{\omega }{3} \right)^{- \frac{3}{2}}} \right)}},{\pi = {3.14.}}}\mspace{14mu} \right.} & (11) \end{matrix}$

The Equation (9) can be linearized by one-term Taylor series expansion:

$\begin{matrix} {{{L\left( {\lambda,\alpha,\beta} \right)} \approx {{\frac{1}{2n}{\sum\limits_{i = 1}^{n}{\left( {Z_{i} - {X_{i}\beta}} \right)^{\prime}{W_{i}\left( {Z_{i} - {X_{i}\beta}} \right)}}}} + {\lambda \left( {{\alpha {\beta }^{1/2}} + {\left( {1 - \alpha} \right){\beta }^{2}}} \right)}}}\mspace{20mu} {{{where}\mspace{14mu} Z_{i}} = {{X_{i}\overset{\sim}{\beta}} + \frac{y_{i} - {f\left( {X_{i}\overset{\sim}{\beta}} \right)}}{{f\left( {X_{i}\overset{\sim}{\beta}} \right)}\left( {1 - {f\left( {X_{i}\overset{\sim}{\beta}} \right)}} \right.}}}} & (12) \end{matrix}$

is the estimated response, W_(i)=f(X_(i){tilde over (β)})(1−f(X_(i){tilde over (β)}) is the weight for the estimated response. f(X_(i){tilde over (β)})=exp(X_(i){tilde over (β)})/(1+exp(X_(i){tilde over (β)})) is evaluated value under the current parameters. Thus, we can redefine the partial residual for fitting current {tilde over (β)} as {hacek over (Z)}_(i) ^((j))=Σ_(k≠j)x_(ik){tilde over (β)}_(k) and ω_(j)=Σ_(i=1) ^(n)W_(i)x_(ij)(Z_(i)−{hacek over (Z)}_(i) ^((j))). A procedure of the coordinate descent method for the HLX penalized logistic model is described as follows.

A coordinate descent method for the HLX penalized logistic model:

-   -   Step 1: Initialize all β_(j)(m)←0(j=1, 2, . . . , p) and X, y,         set m←0, λ and α are chosen by cross-validation;     -   Step 2: Calculate Z(m) and W(m) and approximate the loss         function (12) based on the current β(m);     -   Step 3: Update each β_(j)(m), and cycle over j=1, . . . , p     -   Step 3.1: Compute {hacek over (Z)}_(i)         ^((j))(m)←Σ_(k≠j)x_(ik)β_(k)(m)     -   and ω_(j)(m)←Σ_(i=1) ^(n)W_(i)(m)x_(ij)(Z_(i)(m)−{hacek over         (Z)}_(i) ^((j))(m));     -   Step 3.2: Update

$\left. {\beta_{j}(m)}\leftarrow\frac{{Half}\mspace{14mu} \left( {{\omega_{j}(m)},{\lambda\alpha}} \right)}{1 + {\lambda \left( {1 - \alpha} \right)}} \right.;$

-   -   Step 4: Let m←m+1, β(m+1)←β(m);     -   If β(m) dose not convergence, then repeat Steps 2, 3

Although not required, the embodiments described with reference to the Figures can be implemented as an application programming interface (API) or as a series of libraries for use by a developer or can be included within another software application, such as a terminal or personal computer operating system or a portable computing device operating system. Generally, as program modules include routines, programs, objects, components and data files assisting in the performance of particular functions, the skilled person will understand that the functionality of the software application may be distributed across a number of routines, objects or components to achieve the same functionality desired herein.

It will also be appreciated that where the methods and systems of the present invention are either wholly implemented by computing system or partly implemented by computing systems then any appropriate computing system architecture may be utilized. This will include standalone computers, network computers and dedicated hardware devices. Where the terms “computing system” and “computing device” are used, these terms are intended to cover any appropriate arrangement of computer hardware capable of implementing the function described.

It will be appreciated by persons skilled in the art that the term “database” may include any form of organized or unorganized data storage devices implemented in either software, hardware or a combination of both which are able to implement the function described.

It will be appreciated by persons skilled in the art that numerous variations and/or modifications may be made to the invention as shown in the specific embodiments without departing from the spirit or scope of the invention as broadly described. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive.

Any reference to prior art contained herein is not to be taken as an admission that the information is common general knowledge, unless otherwise indicated.

EXAMPLES Example 1 Analyses of Simulated Data

The goal was to evaluate the performance of logistic regression with the HLX regularization in the simulation study.

Four approaches were compared with the approach according to the present invention: logistic regression with the Lasso regularization, L_(1/2) regularization, SCAD-L₂ and Elastic net regularization respectively. The data have been simulated from the true model

${{\log \left( \frac{y}{1 - y} \right)} = {{X\; \beta} + {\sigma\varepsilon}}},{\varepsilon \text{∼}{{N\left( {0,1} \right)}.}}$

Where X˜N(0,1), ε is the independent random error and σ is the parameter which controls the signal to noise. Three scenarios are presented here. In every example, the dimension of predictors was 2000. The notation ./. represents the number of observations in the training and test sets respectively, e.g. 100/100. Here are the details of the three scenarios.

(a) In scenario 1, the dataset consisted of 100/100 observations, σ was set=0.3 and

$\beta = {\left( {\underset{\underset{5}{}}{2,2,2,2,2},\underset{\underset{1995}{}}{0,\ldots \mspace{11mu},0}} \right).}$

A grouped variable situation has been simulated

x _(i) =ρ×x ₁+(1−ρ)×x _(i) , i=2, 3, 4, 5.

where ρ is the correlation coefficient of the grouped variables.

(b) The scenario 2 was defined similarly to scenario 1, except that the case when there are other independently factors also contributes to the y has been considered

$\beta = {\left( {\underset{\underset{5}{}}{2,2,2,2,2},\underset{\underset{5}{}}{0,\ldots \mspace{11mu},0},\underset{\underset{5}{}}{1.5,{- 2},1.7,3,{- 2.5}},\underset{\underset{1985}{}}{0,\ldots \mspace{11mu},0}} \right).}$

(c) In scenario 3, σ was set=0.4 and the dataset consisted of 200/200 observations, and

$\beta = {\left( {\underset{\underset{5}{}}{2,2,2,2,2},\underset{\underset{5}{}}{0,\ldots \mspace{11mu},0},\underset{\underset{5}{}}{1.5,{- 2},1.7,3,{- 2.5}},\underset{\underset{5}{}}{\quad{0,\ldots \mspace{11mu},0}},\underset{\underset{5}{}}{3,3,3,3,3},\underset{\underset{5}{}}{0,\ldots \mspace{11mu},0},\underset{\underset{5}{}}{3,3,3,{- 3},{- 2}},\underset{\underset{1965}{}}{0,\ldots \mspace{11mu},0}} \right).}$

Two grouped variables have been defined

x _(i) =ρ×x ₁+(1−ρ)×x _(i) , i=2, 3, 4, 5;

x _(i) =ρ×x ₂₁+(1−ρ)×x _(i) , i=22, 23, 24, 25.

In this example, there were four important groups, and within each group there have been five members, correlated or independent. An ideal method would select only the 20 true features and set the coefficients of the 1980 noise features to zero.

In this example, correlation coefficient was set ρ=0.3, 0.6, 0.9 respectively. The Lasso and Elastic net were conducted by Glmnet (a Matlab package, version 2014-04-28, download at http://web.stanford.edu/˜hastie/glmnet_matlab/). The regularization parameters of the Lasso, L_(1/2), SCAD-L₂, Elastic net and the HLX approaches were tuned by 10-fold cross validation (CV) in the training set. The Elastic net and HLX methods were tuned by 10-CV on the two dimensional parameter surfaces. The SCAD-L₂ has three parameters (α, λ₁ and λ₂), and α was set=2, 2.25, 2.5, 3, 3.5, 3.7, respectively, λ₁ and λ₂ were also tuned by 10-CV on the two dimensional parameter surfaces.

The simulations were repeated 100 times and the classification accuracy was computed on the test sets. To evaluate the quality of the selected features, the sensitivity and specificity of variable selection performance (Zhang W. et al., BMC Genomics. England; 2013;14 Suppl 8: S7-2164-14-S8-S7. Epub 2013 Dec. 9. doi:10.1186/1471-2164-14-S8-S7 [doi]) was defined as follows:

-   -   True Negative (TN) :=|β.*{circumflex over (β)}|₀, False Positive         (FP) :=|β.*{circumflex over (β)}|₀     -   False Negative (FN) :=|β.*{circumflex over (β)}|₀, True Positive         (TP) :=|β.*{circumflex over (β)}|₀

${{Sensitivity}:=\frac{TP}{{TP} + {FN}}},{{Specificity}:={\frac{TN}{{TN} + {FP}}.}}$

where the .* is the element-wise product and |.|₀ calculates the number of non-zero elements in a vector, β is the logical “not” operator on a vector.

For all scenarios, in general, the HLX procedure proposed with the present invention and embodiments thereof gave higher or comparable accuracy than the Lasso, SCAD-L₂, Elastic net and L_(1/2) methods (Table 1). In addition, the HLX penalty resulted in much higher sensitivity in identifying true features compared to the other four methods. For example, in the scenario 1 with p=0.9, the method of the present invention gained impressive performance (Accuracy 99.63% with perfect Sensitivity and perfect Specificity). The specificity was somewhat decreased, but not greatly as compared to the achieved in sensitivity.

TABLE 1 Results of the simulation. Scenario 1 2 3 1 2 3 1 2 3 p Methods Sensitivity Specificity Accuracy 0.3 Lasso 0.948 0.540 0.238 0.995 0.989 0.991 91.40% 68.90% 65.26% L_(1/2) 1.000 0.700 0.655 0.998 0.997 0.975 92.00% 73.40% 70.40% SCAD-L₂ 1.000 0.757 0.600 0.999 0.971 0.923 92.70% 72.90% 67.51% EN 0.973 0.772 0.740 0.994 0.975 0.844 90.00% 70.40% 64.47% HLX 1.000 0.792 0.793 0.999 0.985 0.934 93.53% 75.54% 68.20% 0.6 Lasso 0.724 0.453 0.280 0.997 0.995 0.993 95.50% 77.43% 74.20% L_(1/2) 0.400 0.400 0.440 0.999 0.997 0.976 96.30% 78.62% 75.30% SCAD-L₂ 1.000 0.770 0.620 1.000 0.960 0.948 98.90% 79.50% 77.20% EN 1.000 0.720 0.766 0.988 0.979 0.861 97.20% 74.53% 69.20% HLX 1.000 0.802 0.698 1.000 0.989 0.947 99.30% 82.68% 76.30% 0.9 Lasso 0.436 0.276 0.173 0.999 0.996 0.995 98.32% 85.16% 77.66% L_(1/2) 0.200 0.400 0.360 1.000 0.998 0.996 98.33% 88.01% 79.00% SCAD-L₂ 1.000 0.690 0.675 0.999 0.984 0.918 99.60% 82.70% 80.60% EN 0.995 0.810 0.731 0.999 0.978 0.877 98.80% 83.90% 74.70% HLX 1.000 0.830 0.745 1.000 0.980 0.911 99.63% 82.60% 85.70% In bold - the best (or one of the best) performance

Example 2 Analyses of Real Data

Three publicly available datasets have been used: Prostate, DLBCL and Lung cancer, to further evaluate the effectiveness of the method of the present invention. The Prostate, DLBCL datasets were downloaded from http://ico2s.org/datasets/microarray.html, and the Lung cancer dataset can be downloaded at www.ncbi.nlm.nih.gov/geo with access number [GSE40419]. A brief description of these three datasets is given below and summarized in Table 2.

Prostate: This dataset was originally proposed by Singh et al. (Singh D, et al., Cancer Cell. Elsevier; 2002;1: 203-209); it is containing the expression profiles of 12,600 genes for 50 normal tissues and 52 prostate tumor tissues. Glaab E. et al. (PLoS One. Public Library of Science; 2012;7: e39932) has been followed to preprocess the data.

Lymphoma: This dataset (Shipp et al. (Shipp M A. et al., Nat Med. Nature Publishing Group; 2002;8: 68-74) contains 77 microarray gene expression profiles of the 2 most prevalent adult lymphoid malignancies: 58 samples of diffuse large B-cell lymphomas (DLBCL) and 19 follicular lymphomas (FL). The original data contains 7, 129 gene expression values. Glaab E. et al. (PLoS One. Public Library of Science; 2012;7: e39932) has been followed to preprocess the data.

Lung cancer: As RNA-sequencing (RNA-seq) technique widely used, therefore, it is important to test the proposed method whether it has the ability to handle the RNA-seq data. To verify it, one dataset which used the next-generation sequencing was involved in our analysis. This dataset (Seo J S. et al., Genome Res. Genomic Medicine Institute (GMI), Medical Research Center, Seoul National University, Seoul 110-799, Korea; 2012;22: 2109-2119. doi:10.1101/gr.145144.112 [doi]) contains 164 samples with 87 lung adenocarcinomas and 77 adjacent normal tissues. We followed Seo JS. et al. (Genome Res. Genomic Medicine Institute (GMI), Medical Research Center, Seoul National University, Seoul 110-799, Korea; 2012;22: 2109-2119. doi:10.1101/gr.145144.112 [doi]) to preprocess the data.

TABLE 2 Datasets used in this example. Dataset No. of samples No. of genes Classes Prostate 102 12600 Normal/Tumor Lymphoma 77 7129 DLBCL/FL Lung cancer 164 22401 Normal/Tumor

Each of three datasets was randomly partitioned into training set (70%˜80%) and test set (20%˜30%). More information on these datasets is given in Table 3.

TABLE 3 Three gene expression datasets used in this example. Dataset No. of Training No. of Testing Prostate 77 25 Lymphoma 57 20 Lung cancer 124 40

The training and the test classification accuracy has been calculated. The performance of area under the curve (AUC) based on receiver operating characteristic (ROC) analysis for the test set was also recorded. The parameter(s) of the regularization methods were tuning by 10-fold CV in one or two dimensions. This procedure has been repeated 50 times, the averaged classification results (training accuracy based on 10-fold CV and accuracy in test set) were summarized in Table 4 and the averaged ROC analysis results are shown in FIG. 5.

As shown in Table 4, for Prostate dataset, the classifier with the HLX regularization gave an average 10-fold CV accuracy of 97.50% and an average test accuracy of 93.88% with about 11 genes selected. The classifiers with Lasso, L_(1/2), SCAD-L₂ and Elastic net methods gave an average 10-fold CV accuracy of 96.09%, 96%, 96.09%, 96.25% and an average test accuracy of 92.08%, 92.08%, 91.31%, 92.40% with about 12, 7, 21 and 15 genes selected respectively. For lymphoma and lung cancer datasets, in general, it can be seen the HLX penalty achieved the best classification performances with the highest accuracy rates in the training and test sets.

As shown in FIG. 5, with a method of the present invention the best classification performances in these three datasets amongst all the competitors can be achieved. For example, the AUC from ROC analysis of the HLX method for datasets prostate, lymphoma and lung cancer were estimated to be 0.9383, 0.9314 and 0.9928 respectively. AUC results of the Lasso method for the three datasets were calculated to be 0.9315, 0.9263 and 0.9809 respectively, which were worse than with a method of the present invention.

TABLE 4 Result of empirical datasets. In bold - the best performance No. of Training accuracy Accuracy selected Dataset Method (10-CV) (testing) genes Prostate Lasso 96.09% 92.08% 12 L_(1/2) 96.00% 92.08% 7 SCAD-L₂ 96.09% 91.31% 21 ElasticNet 96.25% 92.40% 15 HLX 97.50% 93.88% 11 Lymphoma Lasso 96.33% 91.00% 18 L_(1/2) 95.25% 91.40% 14 SCAD-L₂ 96.10% 93.67% 24 ElasticNet 95.98% 92.00% 31 HLX 96.39% 93.83% 16 Lung Lasso 97.46% 98.33% 10 cancer L_(1/2) 98.38% 97.50% 9 SCAD-L₂ 97.85% 98.29% 25 ElasticNet 97.18% 98.38% 24 HLX 98.63% 98.33% 15

The top 10 ranked genes selected by the four regularization methods for the lung cancer gene expression dataset are summarized in Table 5, the information of top 10 ranked genes for the other datasets are presented in Tables 6 and 7.

Considering that a method of the present invention allowed for the best performances to select the true features in the simulation (Table 1), it is implied that the genes selected by the HLX method in these three cancer datasets are valuable to the researchers who the goal is to find out the key factors that associated with the cancer development. For example, in Table 5, the biomarkers selected with a method of the present invention include advanced glycosylation end product receptor (AGER), which is a member of the immunoglobulin superfamily predominantly expressed in the lung. AGER plays a role in epithelial organization, and decreased exxression of AGER in lung tumors may conduce to loss of epithelial tissue structure, potentially leading to malignant transformation (Bartling B. et al., Carcinogenesis. Clinic of Cardiothoracic Surgery, Martin Luther University Halle-Wittenberg, Ernst-Grube-Strasse 40, D-06120 Halle/Saale, Germany; 2005;26: 293-301. doi:bgh333 [pii]). The unique function of AGER in lung indicate that it could be used as an additional diagnostic tool for lung cancer (Buckley S T, Ehrhardt C., J Biomed Biotechnol. School of Pharmacy and Pharmaceutical Sciences, Trinity College Dublin, Dublin 2, Ireland.; 2010;2010: 917108. doi:10.1155/2010/917108 [doi]), and even as a target (Jing R. et al., Neoplasma. Center of Laboratory Medicine, Affiliated Hospital of Nantong University, Affiliated Hospital of Nantong University, 20 Xi Si Road, Nantong 226001, PR China; 2010;57: 55-61). Neutrophil cytosolic factor 1 (NCF1), the regulatory subunit of NADPH oxidase:

The expression profile of NCF1 was associated with lung cancer development (Kelkka T. et al., PLoS One. Public Library of Science; 2013;8:e84148.doi:10.1371/journal.pone.0084148, Vuong H. et al., Genome Med. Department of Biomedical Informatics, Vanderbilt University School of Medicine, 2525 West End Avenue, Suite 600, Nashville, Tenn. 37203 USA; Department of Biomedical Informatics, Vanderbilt University School of Medicine, 2525 West End Avenue, Suit (2014;6: 81-014-0081-7. eCollection 2014. doi:10.1186/s13073-014-0081-7 [doi])).

Moreover, interestingly, there were at least two groups within the 10 top-rank genes selected by HLX. For example, the correlation coefficients between the expression of AGER and ABCA8, EIF4E1B and CXXC1P1 were calculated to be 0.5 and 0.8, respectively. They were successfully selected as group variables with a method of the present invention.

TABLE 5 The 10 top-ranked informative genes found by the five sparse logistic regression methods from the Lung cancer dataset. Rank Lasso L_(1/2) SCAD-L₂ ElasticNet HLX 1 KRTAP4-6 A2M SCUBE1 ERI2 AGER 2 FAM150B A4GNT MIR548Y C16orf70 ABCA8 3 RADIL ABCA8 CAV1 AGER LGI3 4 MIR548Y AATK C15orf32 IL1F9 MIR548Y 5 PHYHIPL B9D2 ID3 CBLN4 MIR98 6 LGI3 BDKRB1 TNXB KRTAP4-6 NCF1 7 AKAP2 PNLIP PHYHIPL SIK2 C10orf54 8 ID3 CDON KCNT2 RADIL EIF4E1B 9 SCUBE1 CASQ1 CCL19 FABP5 CXXC1P1 10 TNXB CD244 ARHGAP44 CFD CYP17A1

TABLE 6 The most frequently selected 10 genes found by the five sparse logistic regression methods from the prostate dataset. Rank Lasso L_(1/2) SCAD-L₂ ElasticNet HLX 1 PTDSS1 PTGDS ATP5I PTDSS1 HPN 2 ATP5I JUNB TP63 G0S2 DUSP1 3 S100A4 RP11-124D2.7 PTDSS1 S100A4 NTRK1 4 AGR2 POLR2M AGR2 TP63 GSTA2 5 XBP1 CYBA MYOF XBP1 TP63 6 HPN RRAD D4S234E JUNB NELL2 7 PTGDS NR4A3 S100A4 C1QTNF3- PTDSS1 AMACR 8 SDC1 CTSG PRKCB FP236383.12 SERINC5 9 JUNB PRKCB CTSG SERPINA3 PHLDA2 10 CFD ATP5I ME1 JUND ATP5I

TABLE 7 The most frequently selected 10 genes found by the five sparse logistic regression methods from the lymphoma dataset. Rank Lasso L_(1/2) SCAD-L₂ ElasticNet HLX 1 TLN2 TLN2 ESPL1 GANAB UPK2 2 ARID4B IFNA2 UPK2 NUP93 TLN2 3 BRD2 IGLV4-3 IFFO1 MIR4668 CCL21 4 IFFO1 CSTA PRDX6 UPK2 RRS1 5 RPL24 DLGAP5 IFNA1 OMD IFFO1 6 PRDX6 CCL21 PNP TLN2 CIRBP 7 CCL21 IFNA1 IGHEP1 CCL21 PSMA4 8 RP11-419C5.2 BRD2 CCL21 CYP4F3 CCR1 9 UPK2 ADIPOQ KDR RPL24 RPL24 10 CIRBP ESPL1 DTYMK PRDX6 GANAB

In addition to comparing with the Lasso, L_(1/2), SCAD-L₂ and Elastic net techniques, a comparison with the results of other methods published in the literature has been consucted. Only methods were considered using CV for evaluation, since approaches based on a mere training/test set partition are now widely known as unreliable (Ambroise C, McLachlan G J., Proc Natl Acad Sci USA. Laboratoire Heudiasyc, Unite Mixte de Recherche/Centre National de la Recherche Scientifique 6599, 60200 Compiegne, France.; 2002;99: 6562-6566. doi:10.1073/pnas.102102699 [doi]).

Table 6 displays the best classification accuracy of other methods. In Table 6, classification accuracy achieved by the HLX regularization approach is greater than other methods meanwhile the number of selected genes is significantly smaller than other methods except on the Lymphoma dataset.

TABLE 6 Result of the literatures. In bold - the best performance* No. of selected Dataset Authors Accuracy (CV) features Prostate T.K. Paul et al. [33] 96.60%   48.5 Wessels et al. [34] 93.40% 14 Shen et al. [35] 94.60% unknown Lecocke et al. [36] 90.10% unknown Dagliyan et al. [37] 94.80% unknown Glaab et al. [24] 94.00% 30 HLX 97.50% 11 Lymphoma Wessels et al. [34] 95.70% 80 Liu et al. [38] 93.50%  6 Shipp et al. [25] 92.20% 30 Goh et al. [39] 91.00% 10 Lecocke et al. [36] 90.20% unknown Hu et al. [40] 87.01% unknown Dagliyan et al. [37] 92.25% unknown Glaab et al. [24] 95.00% 30 HLX 96.39% 16 *[33] Paul T K, Iba H., ACM; 2005. pp. 453-460, [34] Wessels L F. et al., Bioinformatics. Department of Mediamatics, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology Mekelweg 4, 2628 CD Delft, The Netherlands; 2005; 21: 3755-3762. doi: bti429 [pii], [35] Shen L., Tan E C., IEEE/ACM Trans Comput Biol Bioinforma. IEEE Computer Society Press; 2005; 2: 166-175, [36] Lecocke M., Hess K., Cancer Inform. Department of Mathematics, St. Mary's University, San Antonio, Texas 78228, USA.; 2007; 2: 313-327, [37] Dagliyan O. et al., PLoS One. Department of Chemical and Biological Engineering, Koc University, Istanbul, Turkey.; 2011; 6: e14579. doi: 10.1371/journal.pone.0014579 [doi], [38] Liu J., Zhou H., Machine Learning and Cybernetics, 2003 International Conference on. IEEE; 2003. pp. 2275-2280, [39] Goh L. et al., Proceedings of the second conference on Asia-Pacific bioinformatics-Volume 29. Australian Computer Society, Inc.; 2004. pp. 161-166, [40] Hu Y., Kasabov N., Neural Information Processing. Springer; 2008. pp. 846-855. 

1. A method for determining an association of one or more biological features with a medical condition, comprising the steps of: obtaining a dataset comprising biological data related to a plurality of samples each having a plurality of biological features; applying at least some of the biological data to a regression model to determine and/or optimize parameters in the regression model thereby solving the regression model; and processing the biological data using the solved regression model to determine one or more biological features that are associated with the medical condition; wherein the regression model involves the use of both L_(1/2)-regularized logistic regression and L₂-regularized logistic regression.
 2. The method in accordance with claim 1, wherein the regression model involves the use of a linear combination of L_(1/2)-regularized logistic regression and L₂-regularized logistic regression.
 3. The method in accordance with claim 1, wherein the dataset is represented as: D={(X ₁ , y ₁), (X ₂ , y ₂), . . . , (X _(n) , y _(n))}, where n is a number of the plurality of samples, p is a number of the plurality of biological features, X_(i)=(x_(i1),x_(i2), . . . ,x_(ip)) is the i^(th) sample with p biological features, and y_(i) is a corresponding dependent variable.
 4. The method in accordance with claim 3, wherein the regression model is defined at least partly by: ${L\left( {\lambda,\alpha,\beta} \right)} = {{- {\sum\limits_{i = 1}^{n}\left\{ {{y_{i}{\log \left\lbrack {f\left( {X_{i}^{\prime}\beta} \right)} \right\rbrack}} + {\left( {1 - y_{i}} \right){\log \left\lbrack {1 - {f\left( {X_{i}^{\prime}\beta} \right)}} \right\rbrack}}} \right\}}} + {\lambda \left( {{\alpha {\beta }^{\frac{1}{2}}} + {\left( {1 - \alpha} \right){\beta }^{2}}} \right)}}$ where β=(β₁, . . . , β_(p)) are the coefficients to be determined and/or optimized, and λ and α are predetermined values.
 5. The method in accordance with claim 4, wherein λ=1 and α=0.2.
 6. The method in accordance with claim 4, wherein the parameters in the regression model are determined and/or optimized using a coordinate descent optimization method.
 7. The method in accordance with claim 6, wherein the coordinate descent optimization method comprises the steps of: optimizing the regression model using at least some of the biological data, by varying one of the parameters in the regression model while fixing the other parameters in the regression model; updating the parameters in the regression model; and repeating the optimization and update steps for at least once for each of the parameters in the regression model so as to optimize the regression model.
 8. The method in accordance with claim 7, wherein the repeating step continues until a convergence condition relating to the parameters in the regression model is reached.
 9. The method in accordance with claim 1, further comprising the step of: ranking the biological features determined to be associated with the medical condition according to their degree of association with the medical condition.
 10. The method in accordance with claim 1, wherein the plurality of samples are taken from humans at least some of which have the medical condition; the plurality of biological features are one or more of : presence of a gene, gene expression, presence of a gene product or amount of a gene product, and the medical condition is cancer.
 11. A system for determining an association of one or more biological features with a medical condition, comprising: one or more processors arranged to apply at least some biological data in a dataset comprising biological data related to a plurality of samples each having a plurality of biological features to a regression model so as to determine and/or optimize parameters in the regression model thereby solving the regression model; and process the biological data using the solved regression model so as to determine one or more biological features that are associated with the medical condition; wherein the regression model involves the use of both L_(1/2)-regularized logistic regression and L₂-regularized logistic regression.
 12. The system in accordance with claim 11, wherein the regression model involves the use of a linear combination of L_(1/2)-regularized logistic regression and L₂-regularized logistic regression.
 13. The system in accordance with claim 11, wherein the dataset is represented as: D={(X ₁ , y ₁), (X ₂ , y ₂), . . . , (X _(n) , y _(n))}, where n is a number of the plurality of samples, p is a number of the plurality of biological features, X_(i)=(x_(i1),x_(i2), . . . ,x_(ip)) is the i^(th) sample with p biological features, and y_(i) is a corresponding dependent variable.
 14. The system in accordance with claim 13, wherein the regression model is defined at least partly by: ${L\left( {\lambda,\alpha,\beta} \right)} = {{- {\sum\limits_{i = 1}^{n}\left\{ {{y_{i}{\log \left\lbrack {f\left( {X_{i}^{\prime}\beta} \right)} \right\rbrack}} + {\left( {1 - y_{i}} \right){\log \left\lbrack {1 - {f\left( {X_{i}^{\prime}\beta} \right)}} \right\rbrack}}} \right\}}} + {\lambda \left( {{\alpha {\beta }^{\frac{1}{2}}} + {\left( {1 - \alpha} \right){\beta }^{2}}} \right)}}$ where β=(β₁, . . . , β_(p)) are the coefficients to be optimized, and λ and α are predetermined values.
 15. The system in accordance with claim 14, wherein λ=1 and α=0.2.
 16. The system in accordance with claim 14, wherein the one or more processors are further arranged to determine and/or optimize the parameters in the regression model by optimizing the regression model using at least some of the data, through varying one of the parameters in the regression model while fixing the other parameters in the regression model; updating the parameters in the regression model; and repeating the optimization and update steps for at least once for each of the parameters in the regression model until a convergence condition relating to the parameters in the regression model is reached.
 17. The system in accordance with claim 11, wherein the one or more processors are further arranged to rank the biological features determined to be associated with the medical condition according to their degree of association with the medical condition.
 18. The system in accordance with claim 11, wherein the plurality of samples are taken from humans at least some of which have the medical condition; the plurality of biological features are one or more of: presence of a gene, gene expression, presence of a gene product or amount of a gene product, and the medical condition is cancer.
 19. The system in accordance with claim 11, further comprising a display arranged to display the determination result.
 20. A non-transient computer readable medium for storing computer instructions that, when executed by one or more processors, causes the one or more processors to perform a method for determining an association of one or more biological features with a medical condition, comprising the steps of: applying at least some biological data in a dataset comprising biological data related to a plurality of samples each having a plurality of biological features to a regression model so as to determine and/or optimize parameters in the regression model thereby solving the regression model; and processing the biological data using the solved regression model so as to determine one or more biological features that are associated with the medical condition; wherein the regression model involves the use of both L_(1/2)-regularized logistic regression and L₂-regularized logistic regression. 